Note that this implies that so-called vertex-frustration [1]}, that is the inability to arrange collectively all vertices in a lowest energy configuration, cannot exist in a graph spin ice whose Hamiltonian depends only on the vertex charge. Indeed, all the vertex-frustrated systems [1]}, many of which have been realized [3]}, [4]} and depend upon a lifting of degeneracy within vertices of the same charge. They are therefore not pure spin ice graphs.
